The a2 + b2 + c2 formula is used to find the sum of squares of three numbers without actually calculating the squares. It says a2 + b2 + c2 = (a + b + c)2 - 2ab - 2bc - 2ca.
a2 + b2 + c2 formula is one of the major algebraic identities. To derive the expansion of a2 + b2 + c2 formula evaluate the (a + b + c)2 formula. Let us learn more about the a2 + b2 + c2 formula along with solved examples.
What is a^2 + b^2 + c^2 Formula?
a2 + b2 + c2 formula says (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca. It is derived from (a + b + c)2 formula as follows.
We just read that by multiplying (a + b + c) by itself we can easily derive the a2 + b2 + c2 formula.
(a + b + c)2 = (a + b + c)(a + b + c)
(a + b + c)2 = a2 + ab + ac + ab + b2 + bc + ca + bc + c2
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
On subtracting 2ab + 2bc + 2ca from both sides of the above formula, the a2 + b2 + c2 formula is:
a2 + b2 + c2 = (a + b + c)2 - 2 (ab + bc + ca)
(or)
a2 + b2 + c2 = (a + b + c)2 - 2ab - 2bc - 2ca
a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca)
This is the expansion of a2 + b2 + c2 formula.
Similarly, we can also express a2 + b2 + c2 formula in one of the following ways:
- a2 + b2 + c2 = (a - b - c)2 + 2ab + 2ac - 2bc
- a2 + b2 + c2 = (a - b + c)2 + 2ab - 2ac + 2bc
- a2 + b2 + c2 = (a + b - c)2 - 2ab + 2ac + 2bc, etc.
Let us see how to use the a2 + b2 + c2 formula in the following section.
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Examples on a2 + b2 + c2 Formula
Let us take a look at a few examples to better understand the formula of a squared plus b squared plus c squared.
Example 1: Find the value of a2 + b2 + c2 if a + b + c = 10 and ab + bc + ca = -2.
Solution:
To find: a2 + b2 + c2
Given that:
a + b + c = 10
ab + bc + ca = -2
Using the a2 + b2 + c2 formula,
a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca)
a2 + b2 + c2 = (10)2 - 2(-2) = 100 + 4 = 104
Answer: ∴ a2 + b2 + c2 = 104.
Example 2: Find the value of a2 + b2 + c2 if a + b + c = -3, 1/a + 1/b + 1/c = -2 and abc = 3.
Solution:
To find: a2 + b2 + c2
Given that:
a + b + c = -3 ... (1)
1/a + 1/b + 1/c = -2 ... (2)
abc = 3 ... (3)
Multiplying (2) and (3),
abc(1/a + 1/b + 1/c) = (3)(−2)
bc + ca + ab = −6
Using the a2 + b2 + c2 formula,
a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca)
a2 + b2 + c2 = (-3)2 - 2(-6) = 9 + 12 = 21
Answer: ∴ a2 + b2 + c2 = 21.
Example 3: Find the value of a2 + b2 + c2 if a + b + c = 20 and ab + bc + ca = 100.
Solution:
To find: a2 + b2 + c2
Given that:
a + b + c = 20
ab + bc + ca = 100
Using the a2 + b2 + c2 formula,
a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca)
a2 + b2 + c2 = (20)2 - 2(100) = 400 - 200 = 200
Answer: ∴ a2 + b2 + c2 = 200.
FAQs on a2 + b2 + c2 Formulas
What is the Expansion of a^2 + b^2 + c^2 Formula?
a2 + b2 + c2 formula is read as a square plus b square plus c square. Its expansion is a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca).
How to Derive a Square Plus b Square Plus c Square Formula Using a2 + b2 Formula?
We know that a2 + b2 = (a + b)2 - 2ab. Replacing b with b + c on both sides:
a2 + (b + c)2 = (a + b + c)2 - 2a (b + c)
a2 + (b2 + c2 + 2bc) = (a + b + c)2 - 2ab - 2ac (∵ (a + b)2 = a2 + b2 + 2ab)
a2 + b2 + c2 = (a + b + c)2 - 2ab - 2ac - 2bc
What is the a^2 + b^2 + c^2 Formula in Algebra?
The a2 + b2 + c2 formula is one of the important algebraic identities. It is read as a square plus b square plus c square. Its a2 + b2 + c2 formula is expressed as a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca).
How to Simplify Numbers Using the a2 + b2 + c2 Formula?
Let us understand the use of the a2 + b2 + c2 formula with the help of the following example.
Example: Find the value of (22 + 52 + 32) using the a2 + b2 + c2 formula.
To find: (22 + 52 + 32)
Let us assume that a = 2 and b = 5 and c = 3.
We will substitute these in the formula of (a2 + b2 + c2).
a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca)
= (2 + 5 + 3)2 - 2(2×5 + 5×3 + 3×2)
= 100 - 62 = 38
Answer: (22 + 52 + 32) = 38
How to Use the (a2 + b2 + c2) Formula Give Steps?
The following steps are followed while using (a2 + b2 + c2) formula.
- Firstly observe the pattern of the numbers whether the three numbers have ^2 as individual power or not.
- Write down the formula of (a2 + b2 + c2).
- a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca).
- Substitute the values of a, b and c in the a2 + b2 + c2 formula and simplify.
